sqrtIma

sqrtIma is a term sometimes used informally to refer to the square root of a complex number's imaginary part. It is not a standard mathematical operator and does not appear in canonical mathematics texts, but it can be used as shorthand in programming or notes when focusing on the imaginary component of a complex value.

Definition: For a complex number z = a + bi, the imaginary part is Im(z) = b. sqrtIma(z) is defined as sqrt(b). If b ≥ 0, the result is a nonnegative real number. If b < 0, sqrt(b) is a purely imaginary number, equal to i sqrt(-b), assuming one uses the principal square root on real inputs.

Examples: z = 3 + 4i yields sqrtIma(z) = 2. z = -2 − 9i yields sqrtIma(z) = 3i. In code, sqrtIma(z) can be implemented as sqrt(z.imag) for nonnegative imag parts, or as a complex square root of a real number when imag is negative.

Usage and cautions: This construct ignores the real part of z and does not represent any standard property of z as a whole. Because it is not standardized, it should be clearly documented when used in mathematical notes or software.

See also: Imaginary part, Complex square root, Square root.